Where m and n are Monads. m is a single-pass initialization monad, while n is a multiple-pass working monad. The arrow supports all of the standard accessory arrow typeclasses, including ArrowLoop if n implements MonadFix, and ArrowApply if m and n are the same.

To the best of my imagination, there cannot be a corresponding useful FactoryMonad. I would be extremely interested if anyone can contradict me on that point.

My current interest is in using the FactoryArrow as a replacement for the mealy-style arrows by using IORefs (or potentially even STM transaction variables) to store automaton states instead of unevaluated thunks.